A simple population model is run as a plain counted loop with no check on the size of…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Computational Physics, Ch Numbers a machine can hold, and the two errors that follow

Problem

A simple population model is run as a plain counted loop with no check on the size of the numbers:

N_{k+1} = N_k\,(1+r\Delta t), \qquad r = 50\,\mathrm{s^{-1}},\ \Delta t = 0.5\,\mathrm{s},\ N_0 = 1.0,

so each step multiplies

N

by the fixed factor

1+r\Delta t = 26

. The printed output shows: step

217

N = 1.12\times10^{307}

; step

218

N = \texttt{inf}

; step

219

N = \texttt{inf}

. A later line then computes a growth-rate estimate

R = \frac{N_{219}-N_{218}}{\Delta t}

and reports

R = \texttt{nan}

. ((a)) Verify, using

\log_{10}26

, that step

218

is indeed the first step at which

N

exceeds the largest finite binary64 magnitude,

1.80\times10^{308}

. ((b)) Name the arithmetic operation that produces the first

\texttt{inf}

. ((c)) Name the arithmetic operation that produces the

\texttt{nan}

in the rate calculation.

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